Editing Heptagon (section)

===Construction===
As 7 is a [[Pierpont prime]] but not a [[Fermat prime]], the regular heptagon is not [[Constructible polygon|constructible]] with [[compass and straightedge]] but is constructible with a marked [[ruler]] and compass. It is the smallest regular polygon with this property. This type of construction is called a [[neusis construction]]. It is also constructible with compass, straightedge and [[angle trisector]]. The impossibility of straightedge and compass construction follows from the observation that <math>\scriptstyle {2\cos{\tfrac{2\pi}{7}} \approx 1.247}</math> is a zero of the [[irreducible polynomial|irreducible]] [[cubic function|cubic]] {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}}. Consequently, this polynomial is the [[minimal polynomial (field theory)|minimal polynomial]] of {{nobreak|2cos({{frac|2π|7}}),}} whereas the degree of the minimal polynomial for a [[constructible number]] must be a power of 2.

{| class=wikitable width=640
|[[File:Neusis-heptagon.png|330px]]<br>A ''neusis construction'' of the interior angle in a regular heptagon.
|[[File:01-Siebeneck-Tomahawk-Animation.gif|380px]]<br>An animation from a neusis construction with radius of circumcircle <math>\overline{OA} = 6</math>, according to [[Andrew M. Gleason]]<ref name="Gleason">{{cite journal|last=Gleason|first=Andrew Mattei|title=Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187 |journal=The American Mathematical Monthly|date=March 1988|volume=95|issue=3 |pages=185–194|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#3 |archiveurl=https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#3 |doi= 10.2307/2323624|jstor=2323624 |archivedate=2015-12-19 |url-status=dead}}</ref> based on the [[angle trisection]] by means of the [[Tomahawk_(geometry)|tomahawk]]. This construction relies on the fact that

<math>6\cos\left(\frac{2\pi}{7}\right)=2\sqrt{7}\cos\left(\frac{1}{3}\arctan\left(3\sqrt{3}\right)\right)-1.</math>
|}

[[File:01-Siebeneck-nach Johnson.gif|thumb|left|400px|Heptagon with ''given side length'':<br />
An animation from a [[neusis construction]] with marked ruler, according to David Johnson Leisk ([[Crockett Johnson]]).]]

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